{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Maximum Margin Classifiers - The Support Vector Machine"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>Given a labeled scatter plot, such as the one below, the best way to classify points of different classes is to find a hyper-plane (straight line in the 2-d case) that separates the two classes.</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x108b50450>]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x108a31490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import course_utils as bd\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')\n",
    "reload(bd)\n",
    "%matplotlib inline\n",
    "\n",
    "#generate data\n",
    "m = [[0.25,0.25], [0.75,0.75]]; s = 0.025; n=20\n",
    "X1 = pd.DataFrame(np.random.multivariate_normal([m[0][0], m[0][1]], [[s,0],[0,s]], n), columns=['x1','x2'])\n",
    "X2 = pd.DataFrame(np.random.multivariate_normal([m[1][0], m[1][1]], [[s,0],[0,s]], n), columns=['x1','x2'])\n",
    "\n",
    "plt.figure(figsize = (8, 8))\n",
    "\n",
    "#Plot data and any hyper-plane that separates the data\n",
    "plt.plot(X1['x1'], X1['x2'],'r.')\n",
    "plt.plot(X2['x1'], X2['x2'],'b+')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>The issue with looking for just any separating line is that an infinite number of lines might qualify, even if the space between classes is quite small. The following shows the same data above along with many qualifying lines.</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 1)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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taJGxjc0DtGlT9XGy9nd1OaTmPny9RsXCPFCdXkcxT8lXRO6PpmxPjsMAfCVm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\n90ttV7BAuxK/bcQHJKKoYgz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      "text/plain": [
       "<matplotlib.figure.Figure at 0x108e9f4d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize = (8, 8))\n",
    "ax = fig.add_subplot(111)\n",
    "\n",
    "#Plot data and any hyper-plane that separates the data\n",
    "plt.plot(X1['x1'], X1['x2'],'r.')\n",
    "plt.plot(X2['x1'], X2['x2'],'b+')\n",
    "\n",
    "#We'll use a brute force search for this, but starting from an intelligent starting point\n",
    "success = 0; ws = np.array([1, 1]); t = 1\n",
    "while (success < 10):\n",
    "    #Randomly perturbate the starting point, check to see if it seprates the points\n",
    "    w_i = ws + np.random.normal(0, 1, 2); t_i = t + np.random.normal(0, 1, 1)\n",
    "    if not ((sum((X1.dot(w_i) - t_i)>0)) or (sum((X2.dot(w_i) - t_i)<0))):\n",
    "        bd.plot_dec_line(X1['x1'].min(), X2['x1'].max(), w_i[0], w_i[1],-1*t_i, 'k-', 't')\n",
    "        success += 1\n",
    "\n",
    "ax.set_xlim(0, 1)\n",
    "ax.set_ylim(0, 1)\n",
    "        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Getting the Maximum Margin\n",
    "<p>With so many hyperplane options, we need a principled way to choose an optimal separating line. Intuitively, we want a line that is simultaneously as far away from the positive instances as it is from the negative instances. In a sense, we want it to be in the middle of the empty region separating the classes. We can formalize this by searching for a hyperplane that maximizes the margin between the positive and negative classes.<br><br>\n",
    "\n",
    "<b>Some Definitions</b>\n",
    "<ul>\n",
    "    <li>$W=<W^1,...W^k>$ is a weight vector, $X=<X^1,...X^k>$ is a vector of features </li>\n",
    "    <li>The separating hyperplane is defined by the line: $W \\cdot X+t=0$</li>\n",
    "    <li>The vector $W$ is called the \"normal vector\" to the line, and is perpendicular to it.\n",
    "    <li>The distance between any point $X_i$ and the line is given by: $\\frac{W \\cdot X_i+t}{\\|W\\|_2}$</li>\n",
    "</ul>\n",
    "<br>\n",
    "<i>(How did we get that formula for the line, you ask??</i><br><br>\n",
    "Let $Q$ be a point on the line $W \\cdot X+t=0$, and let $W$ originate from $Q$. The distance from $X_i$ to the line is equal to the projection of the vector $\\vec{QX_i}$ onto $W$. The distance $d$ is given then by the projection formula:<br>\n",
    "<center>$d = \\frac{W \\cdot \\vec{QX_i}}{\\|W\\|_2} = \\frac{W \\cdot X_i - W \\cdot Q}{\\|W\\|_2} = \\frac{W \\cdot X_i+t}{\\|W\\|_2}$ </center><br>\n",
    "\n",
    "The last bit comes from the fact that $W \\cdot Q = t$ since $Q$ is on the line.<br><br>\n",
    "\n",
    "<b>Constraining the Problem</b><br><br>\n",
    "By adding contraints to the problem we make it easier to solve. Let $M^+$ and $M^-$ be the distances from the closest positive and negative point to the line. We are free to rescale $t$, $\\|W\\|_2$, $M^+$ and $M^-$ , so we do the following:<br><br>\n",
    "<center>$M^+ = M^- = m = 1$</center><br><br>\n",
    "With this constraint, the margin then becomes $\\frac{2m}{\\|W\\|_2}$. Maximizing the margin then is equivalent to minimizing $\\|W\\|_2$, or more conveniently, $\\frac{1}{2}\\|W\\|_2$.\n",
    "</p>\n",
    "\n",
    "##Defining the Optimization Problem\n",
    "\n",
    "<p>\n",
    "\n",
    "With perfectly separable data, we then define the following objective function.<br><br>\n",
    "\n",
    "<center>$W^*,t^* = \\underset{W,t} {\\mathrm{argmin}}\\frac{1}{2}\\|W\\|_2^2$  $\\,\\,\\,\\,\\,$   subject to $y_i(W\\cdot X_i + t) \\geq 1$</center>\n",
    "<br>\n",
    "<br>\n",
    "We can transform this using the method of Langrange multipliers. For each constraint we introduce a multipler $\\alpha_i$, which results in the following Lagrange function:<br><br>\n",
    "<center>$\\Lambda(w,t,\\alpha_i,...\\alpha_n)=\\frac{1}{2}\\|W\\|_2^2-\\sum\\limits_{i=1}^n\\alpha_i(y_i(W\\cdot X_i+t)-1)$</center>\n",
    "<br>\n",
    "<br>\n",
    "The optimal hyper-plane is a saddle point of $\\Lambda(w,t,\\alpha_i,...\\alpha_n)$. An interesting consequence of this formula is that there is a \"dual-form,\" where we eliminate $W$ and $t$ and formulate the problem completely in terms of the Lagrange multipliers $\\alpha_i$. The dual-form is:<br><br>\n",
    "\n",
    "<center>$\\underset{\\alpha_1,...,\\alpha_n} {\\mathrm{argmax}}\\sum\\limits_{i=1}^n \\alpha_i - \\frac{1}{2} \\sum\\limits_{i=1}^n \\sum\\limits_{j=1}^n \\alpha_i \\alpha_j y_i y_j X_i\\cdot X_j$\n",
    "</center>\n",
    "<center>subject to $\\alpha_i\\geq 0$ and $\\sum\\limits_{i=1}^n \\alpha_i y_i = 0$</center>\n",
    "<br>\n",
    "<br>\n",
    "The significance of the dual form is that searching for the optimal plane is equivalent to searching for the <i>support vectors</i>, which are the points that lie on the margin planes, and mathematically, are the points with non-zero Langrange multipliers. The decision boundary can be found using:<br><br>\n",
    "<center>$W=\\sum\\limits_{X \\in SV} \\alpha_i y_i X_i$</center>\n",
    "<br>\n",
    "<br>\n",
    "\n",
    "Now let's stop beating around the bush and actually build one! (note one important tip, with SVM's we generally encode a binary outcome as $[-1, 1]$ and not $[0, 1]$).\n",
    "\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Ro0ezjkYeQj1tEyeRSODs7PzoJ69cAdLTgSNHUCuTwdbWFnfu3GETkOiMvb09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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1090e1c10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn import svm\n",
    "import course_utils as bd\n",
    "import pandas as pd\n",
    "reload(bd)\n",
    "\n",
    "m = [[0.25,0.25],[1.75,1.75]]; s = 0.05; n=20\n",
    "X1=pd.DataFrame(np.random.multivariate_normal([m[0][0],m[0][1]],[[s,0],[0,s]],n), columns=['x1','x2'])\n",
    "X2=pd.DataFrame(np.random.multivariate_normal([m[1][0],m[1][1]],[[s,0],[0,s]],n), columns=['x1','x2'])\n",
    "X1['Y'] = -1 * np.ones(X1.shape[0]); X2['Y'] = 1 * np.ones(X2.shape[0])\n",
    "dat = X1.append(X2, ignore_index=True)\n",
    "\n",
    "\n",
    "my_svm = svm.SVC(kernel='linear')\n",
    "my_svm.fit(dat[['x1','x2']], dat['Y'])\n",
    "\n",
    "bd.plotSVM(dat[['x1','x2']], dat['Y'], my_svm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.00000005,  1.00000008])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_svm.support_vectors_.dot(my_svm.coef_[0])+my_svm.intercept_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>In the above we plot the optimal linear hyper-plane $W$ and also the hyper-planes that define the margin. Circled are the support vectors.<br><br>\n",
    "\n",
    "Once we have found $W$ and $t$, classification of a point $X$ is defined as:<br><br>\n",
    "<center>$f(X) = sign(W\\cdot X + t)$</center><br><br>\n",
    "which is just the sign of the distance between $X$ and the optimal hyper-plane.<br><br>\n",
    "\n",
    "In the above scenarios we identified an optimal separating hyper-plane for data that is cleanly separated by a decision boundary. This is seldom the case in real data, so we need to adapt our formulation to allow for such cases.\n",
    "</p>\n",
    "\n",
    "##The Soft Margin Classifier\n",
    "\n",
    "<p>Let $\\xi_i = [\\xi_i]_+ = [1- y_i(W\\cdot X_i+t)]_+$ be the distance between a wrongly classified point $X$ and the optimal margin plane $y_i(W\\cdot X_i+t)=1$, where $[1-z]_+ = max(0,1-z)$. This last function is also known as the <i>hinge function</i>. We can then redefine our optimization problem to allow for some misclassification of our training points. Specifically,:<br><br>\n",
    "\n",
    "<center>$W^*,t^* = \\underset{W,t} {\\mathrm{argmin}}\\frac{1}{2}\\|W\\|_2^2 + C\\sum\\limits_{i=1}^n\\xi_i$</center>  \n",
    "<center>subject to $y_i(W\\cdot X_i + t) \\geq 1 - \\xi_i$</center><br><br>\n",
    "\n",
    "It is often useful to write this as:\n",
    "<center>$W^*,t^* = \\underset{W,t} {\\mathrm{argmin}}\\;\\lambda\\|W\\|_2^2 + \\sum\\limits_{i=1}^n [1- y_i(W\\cdot X_i+t)]_+$</center><br><br>\n",
    "where $\\lambda = \\frac{1}{2C}$. When we view the optimization problem in the above light, we can connect this back to ERM. The second term defines the training error, or the <i>hinge loss</i> of the problem. Our goal is to minimize this loss function while minimizing $W$, which in turn maximizes the margin.\n",
    "\n",
    "\n",
    "</p>\n",
    "\n",
    "##Example Soft Margin\n",
    "\n",
    "<p>When we presented the soft margin version of the SVM, we also introduced a new term $C$ which controls the amount of error we can tolerate (or conversely, the size of the margin). $C$ is thus a free parameter that we must choose to get the right fit. One way to think about it is as $C$ increases we increase the size of the 2nd term in our optimization function. This means we become <i>less</i> tolerant of training errors (and we thus decrease the size of the margin).\n",
    "\n",
    "\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Llai9Bw8ewNjYG0Dpi1x9+/ZF3759MWvWh9vY06dPY9CgQVIdB7sYGk+78mLV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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1095fe290>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn import svm\n",
    "import course_utils as bd\n",
    "import pandas as pd\n",
    "reload(bd)\n",
    "\n",
    "\n",
    "#Generate noisier data than the last example\n",
    "m = [[0.25,0.25],[1.25,1.25]]; s = 0.35; n=20\n",
    "X1=pd.DataFrame(np.random.multivariate_normal([m[0][0],m[0][1]],[[s,0],[0,s]],n), columns=['x1','x2'])\n",
    "X2=pd.DataFrame(np.random.multivariate_normal([m[1][0],m[1][1]],[[s,0],[0,s]],n), columns=['x1','x2'])\n",
    "X1['Y'] = -1 * np.ones(X1.shape[0]); X2['Y'] = 1 * np.ones(X2.shape[0])\n",
    "dat = X1.append(X2, ignore_index=True)\n",
    "\n",
    "\n",
    "#Now plot for different levels of C\n",
    "\n",
    "fig = plt.figure()\n",
    "c_s = [ 1000, 1, 10e-2, 10e-4 ] \n",
    "for i,c in enumerate(c_s):\n",
    "    my_svm = svm.SVC(kernel='linear', C=c)\n",
    "    my_svm.fit(dat[['x1','x2']], dat['Y'])\n",
    "    w = my_svm.coef_[0]\n",
    "    t = my_svm.intercept_\n",
    "    if t<0:\n",
    "        sign = '-'\n",
    "    else:\n",
    "        sign = '+'\n",
    "    ax = fig.add_subplot(2, 2, i+1)\n",
    "    plt.title('C={} : f=<{},{}> X {} {}'.format(c, round(w[0],1), round(w[1],1), sign,np.abs(round(t,1))))\n",
    "    bd.plotSVM(dat[['x1','x2']], dat['Y'], my_svm)\n",
    "    ax.axes.get_xaxis().set_visible(False)\n",
    "    ax.axes.get_yaxis().set_visible(False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>As we increase $C$, we can see that the norm of $W$ increases and the number of support vectors identified shrinks. \n",
    "</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#SVM vs. Logistic Regression\n",
    "\n",
    "<p>Both SVM and Logistic Regression finds linear separating hyper-planes of the form $class(X)=f(W\\cdot X+t)$.\n",
    "\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10b5f9c10>"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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kXTP7+qvr8DjlAWMdY+xw2gFD7UY4ehRYsoRbBPTrr3SHCKkcqB44kTlpJ9OO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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b2d85d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn import svm, linear_model\n",
    "import course_utils as bd\n",
    "from sklearn.metrics import confusion_matrix\n",
    "import pandas as pd\n",
    "reload(bd)\n",
    "\n",
    "#Generate training and testing data from the same distribution\n",
    "m = [[0.25,0.25],[1.25,1.25]]; s = 2; n=20\n",
    "\n",
    "def makeData(m, s, n):\n",
    "    X1=pd.DataFrame(np.random.multivariate_normal([m[0][0],m[0][1]],[[s,0],[0,s]],n), columns=['x1','x2'])\n",
    "    X2=pd.DataFrame(np.random.multivariate_normal([m[1][0],m[1][1]],[[s,0],[0,s]],n), columns=['x1','x2'])\n",
    "    X1['Y'] = -1 * np.ones(X1.shape[0]) \n",
    "    X2['Y'] = 1 * np.ones(X2.shape[0])\n",
    "    return X1.append(X2, ignore_index=True)\n",
    "\n",
    "train = makeData(m, s, n)\n",
    "test = makeData(m, s, n)\n",
    "\n",
    "#Fit both types of models\n",
    "logreg = linear_model.LogisticRegression(C=1e30)\n",
    "logreg.fit(train[['x1','x2']], (train['Y']+1)/2)\n",
    "\n",
    "my_svm = svm.SVC(kernel='linear')\n",
    "my_svm.fit(train[['x1','x2']], train['Y'])\n",
    "\n",
    "#Get accuracy\n",
    "def getACC(X_t, Y_t, mod):\n",
    "    cm=confusion_matrix(mod.predict(X_t),Y_t)\n",
    "    return (cm[0][0]+cm[1][1])/float(sum(cm))\n",
    "\n",
    "acc_svm = getACC(test[['x1','x2']], test['Y'], my_svm)\n",
    "acc_lr = getACC(test[['x1','x2']], (test['Y']+1)/2, logreg)\n",
    "\n",
    "#Plot them\n",
    "w_svm = my_svm.coef_[0]; a_svm = -w_svm[0] / w_svm[1]; t_svm = my_svm.intercept_[0];\n",
    "w_lr = logreg.coef_[0]; a_lr = -w_lr[0] / w_lr[1]; t_lr = logreg.intercept_[0];\n",
    "\n",
    "xx = np.linspace(test[['x1','x2']].iloc[:,0].min(), test[['x1','x2']].iloc[:,1].max())\n",
    "\n",
    "yy_svm = a_svm * xx - (t_svm) / w_svm[1]\n",
    "yy_lr = a_lr * xx - (t_lr) / w_lr[1]\n",
    "\n",
    "\n",
    "plt.plot(test[(test['Y']==-1)].iloc[:,0], test[(test['Y']==-1)].iloc[:,1],'r.')\n",
    "plt.plot(test[(test['Y']==1)].iloc[:,0], test[(test['Y']==1)].iloc[:,1],'b+')\n",
    "plt.plot(xx, yy_svm, label='SVM({})'.format(acc_svm))\n",
    "plt.plot(xx, yy_lr, label='LR({})'.format(acc_svm))\n",
    "plt.legend()"
   ]
  },
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    "<p>We can see that the two classifiers behave very similarly when just looking for a linear decision boundary. The choice of which classifier to use is usually determined by comfort and familiarity as well as the need to exploit the specific properties of each.<br><br>\n",
    "\n",
    "<ul><b>SVM</b>\n",
    "    <li>Intuitive geometric interpretation</li>\n",
    "    <li>Extensibility to non-linear decision surfaces</li>\n",
    "    <li>Strong learning guarantees</li>\n",
    "</ul>\n",
    "<ul><b>Logistic Regression</b>\n",
    "    <li>Designed for estimating $P(Y|X)$</li>\n",
    "    <li>Strong statistical properties</li>\n",
    "    <li>Suited for ranking</li>\n",
    "</ul>\n",
    "</p>"
   ]
  }
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